Averaging a representation over a subgroup

Author:
R. B. Burckel

Journal:
Proc. Amer. Math. Soc. **78** (1980), 399-402

MSC:
Primary 22D10; Secondary 43A07

DOI:
https://doi.org/10.1090/S0002-9939-1980-0553383-3

MathSciNet review:
553383

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Abstract: The purpose of this note is to extend a well-known technique for getting a representation of a quotient group from one of the original group. This is usually done by ``integrating'' coefficient functions of the representation over the subgroup, i.e., by applying some mean to them. Hence amenability hypotheses are usually made. None are needed here because the relevant coefficient functions belong to the algebra of weakly almost periodic functions (Eberlein [**3**]), which is always amenable (Ryll-Nardzewski [**5**]).

**[1]**R. B. Burckel,*Weakly almost periodic functions on semigroups*, Gordon and Breach Science Publishers, New York-London-Paris, 1970. MR**0263963****[2]**Ching Chou,*Uniform closures of Fourier-Stieltjes algebras*, Proc. Amer. Math. Soc.**77**(1979), no. 1, 99–102. MR**539638**, https://doi.org/10.1090/S0002-9939-1979-0539638-9**[3]**W. F. Eberlein,*Abstract ergodic theorems and weak almost periodic functions*, Trans. Amer. Math. Soc.**67**(1949), 217–240. MR**0036455**, https://doi.org/10.1090/S0002-9947-1949-0036455-9**[4]**Pierre Eymard,*L’algèbre de Fourier d’un groupe localement compact*, Bull. Soc. Math. France**92**(1964), 181–236 (French). MR**0228628****[5]**Czesław Ryll-Nardzewski,*On fixed points of semigroups of endomorphisms of linear spaces*, Proc. Fifth Berkeley Sympos. Math. Statist. and Probability (Berkeley, Calif., 1965/66) Univ. California Press, Berkeley, Calif., 1967, pp. 55–61. MR**0215134**

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Additional Information

DOI:
https://doi.org/10.1090/S0002-9939-1980-0553383-3

Keywords:
Unitary representation,
weakly almost periodic function,
mean value for WAP functions,
Fourier-Stieltjes algebra (nonabelian)

Article copyright:
© Copyright 1980
American Mathematical Society